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CMI Entrance 2019 - Answer Key, Sequential Hints

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Please submit problems in the comment section. Try to make the statement as close to the actual statement as possible.

This post contains questions from Chennai Mathematical Institute, CMI Entrance 2019, Sequential hints, answer key, solutions. Our team will upload the solutions. Problems are contributed by students.

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Problems

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1. Suppose \( \frac{8^x + 27^x}{12^x + 18 ^x} = 7/6 \). Find x.

2. There is a parallelogram ABCD. O is any point in the interior such that angle AOB + angle DOC = 180. Prove that angle ODC= angle OBC

3. Find the number of natural number n for which \(n^6+n^4+1 \) is a perfect square

4. \(\phi_k (m) = \phi_1 (\phi_{k-1} (m)) \)  and \(\phi_1 (m) \) is number of divisors of m . Find k for which \( \phi_k (2019^2019) =2 \)

5. If there exists a calculator with 12 buttons, 10 being the buttons for the digits and A and B being two buttons being processes where if n is displayed on the calculator if A is pressed it increases the displayed number by 1 and if B is pressed it multiplies n by 2 hence 2n. Hence find the minimum number of moves to get 260 from 1

6. Find the sum 1+111+11111+1111111+.....1....111(2k+1) ones 

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Objective Section (Answer Key)

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1. 2. 3. 4. 5. 6. 
7. 8. 9. 10. 11. 12. 
13. 14. 15. 16. 17. 18. 
19. 20. 21. 22. 23. 24. 
25.26. 27.28. 29.30. 
 

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Subjective Section (Sequential Hints)

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Faculty Team (for this section)

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Founder - Faculty at Cheenta. Pursuing Ph.D. at University of Wisconsin, Milwaukee. USA Research Interest: Geometric Group Theory

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Faculty - Admin at Cheenta Pursuing B.Stat at Indian Statistical Institute, India

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Faculty at Cheenta Pursuing B.Sc. Math at Chennai Mathematical Institute, India Research Interest - Algebraic Geometry

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Associate Faculty at Cheenta

Pursuing B.Stat from Indian Statistical Institute, India

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